Parameterized complexity of categorical clustering with size constraints

In the Categorical Clustering problem, we are given a set of vectors (matrix) $A=\{a_1,\dots ,a_n\}$ over ${\Sigma }^m$, where Σ is a finite alphabet, and integers k and B. The task is to partition A into k clusters such that the median objective of the clustering in the Hamming norm is at most B. Fomin, Golovach, and Panolan [ICALP 2018] proved that the problem is fixed-parameter tractable for the binary case $\Sigma =\{0,1\}$. We extend this algorithmic result to a popular capacitated clustering model, where in addition the sizes of the clusters are lower and upper bounded by integer parameters p and q, respectively. Our main theorem is that the problem is solvable in time $2^{O(B\log B)}|\Sigma {|}^B\cdot {\left(mn\right)}^{O\left(1\right)}$.